One of the challenges with a real ZX81 is getting it to display on a modern television. Luckily, there are a few simple mods available that will make the ZX81 output a composite video signal. The circuit pictured above is what I used. It's a simple circuit which I found here. An easy mod to make even for a relative newcomer to soldering. If you prefer a more 'off the shelf' solution, a board can be bought here, which is more sophisticated than the circuit I used
Over the years, the most likely thing to fail on a zx81 is the keyboard membrane. The chances are that if you buy a real ZX81, you'll have to replace the keyboard. Again, not the most difficult thing in the world to do. You can get a replacement keyboard from here, as well as full nstructions on how to fit it. The procedure is fairly simple and requires no soldering
A ZX81 is for most purposes useless without a 16K RAM Pack and it is probably cheaper to buy a ZX81 + 16K RAM Pack together than it is to buy a ZX81 on it's own then try and find a working RAM Pack later on.